Have you looked into anything from Buck Knives? They do a great job with thin grinds that work well for hunting applications.
If you're open to handmade knives, there are many options out there. A couple of mine run right around your price range, as a reference point, so you should be able to journey down that path as well. But, it can get a little murky given the overwhelming amount of options.
Regarding the mini finn, I would recommend against it as it has a really small handle. For me, even if a blade is small, I still like a larger handle.
I did a quick look on BladeHQ to see what's currently available as I havne't kept up on smaller blades recently as I've been working on the opposite end of the spectrum the past two years. Here are some that would "speak to me" of looking for a small game and bird knife and is around your price range, that wasn't one of my designs.
Grohmann Canadian belt knife
Esee Xandcudo (A little over your budget, maybe it will be discounted during Black Friday or the holiday season?)
Cold steel mini Pendleton hunter (several options in blade steel at different price points)
Tops bird and trout (kind of a modern take on the Grohmann above)
Buck small pursuit (regular or pro, different color and different blade steel)
Brisa (enzo) necker 70
These are my preferences and not necessarily yours. What I'm looking for, for such a use is a thin and slicey edge (so, not a thick blade), enough handle to stay grippy when wet or bloody, something with a good point for piercing and a little belly but not a lot, and something easy to carry.